The curves $y = \frac { 1 } { 2 } \pi \cos x$ and $x = \frac { 1 } { 2 } \pi \cos y$ intersect at the three points $\left( 0 , \frac { 1 } { 2 } \pi \right) , ( a , b )$, $\left( \frac { 1 } { 2 } \pi , 0 \right)$, as shown in the figure below.\ (a) Explain why $a = b = \frac { 1 } { 2 } \pi \cos b$.\ (b) Show that $\pi \sin b = \sqrt { \pi ^ { 2 } - 4 b ^ { 2 } }$.\ (c) Show that the area of the shaded region is
$$\sqrt { \pi ^ { 2 } - 4 b ^ { 2 } } - \frac { \pi } { 2 } - b ^ { 2 }$$